Surprise surprise, it appears to be Leonard Euler’s “Universal Arithmetic”, written by him in St Petersburg and published there in 1768 in Russian translation produced by his students:

It clearly set out standards of quality of mathematical content and enshrined the propaedeutics principle so visible in the Russian tradition: a textbook was supposed to be a stepping stone to further more advanced study.

I am proud that, after marking 220 examination scripts in first year linear algebra , I was still able to locate, at a glance, an error in this picture — thanks to skills in parsing of meaningless symbolic input developed over many years of teaching mathematics.

The modal view in the cognitive and neural sciences holds that consciousness is necessary for abstract, symbolic, and rule-following computations. Hence, semantic processing of multiple-word expressions, and performing of abstract mathematical computations, are widely believed to require consciousness. We report a series of experiments in which we show that multiple-word verbal expressions can be processed outside conscious awareness and that multistep, effortful arithmetic equations can be solved unconsciously. All experiments used Continuous Flash Suppression to render stimuli invisible for relatively long durations (up to 2,000 ms). Where appropriate, unawareness was verified using both objective and subjective measures. The results show that novel word combinations, in the form of expressions that contain semantic violations, become conscious before expressions that do not contain semantic violations, that the more negative a verbal expression is, the more quickly it becomes conscious, and that subliminal arithmetic equations prime their results. These findings call for a significant update of our view of conscious and unconscious processes.

Today brought big news from Chile regarding my brother’s disappearance. After many many years of frustration, arrest warrants have been issued for 8 police and military officers for the kidnapping and enforced disappearance of my brother, who went missing in 1985.

I refer to my old post Women and mathematics in relation to the caustic comic strip by Zach Weiner. Both my post (which later became a section in my book Mathematics under the Microscope) and Weiner’s cartoons are about the place of a woman in the violent world of mathematics and about men’s perception of women’s place. If you think that this is an exaggeration then read comments to my post, like this one, from a female colleague:

As one of the few female mathematicians in Alexandre’s field I think he is correct.

You probably have to be a research mathematician to understand what he is saying about being bold, needing intellectual independence, the psychologically charged and tense discourse and everyone looking and acting as if they are going to get into a fist fight any moment.

This is typically not how women behave and when a woman does act like this (learned or natural) she gets all sorts of criticism for not being the sweet docile soft person she looks like. And all too often the criticism is from other women, not just from men.

Part of my anger and frustration at school was that so much of this subject
that I loved, mathematics, was wasted on what I thought was frivolous or
immoral applications: frivolous because of all those unrealistic puzzles,
and immoral because of the emphasis on competition (Olympiads, chess, card
games, gambling, etc). I had (and retain) a profound dislike of
competition, and I don’t see why one always had to demonstrate one’s
abilities by beating other people, rather than by collaborating with them.
I believed that “playing music together”, rather than “playing sport against
one another”, was a better metaphor for what I wanted to do in life, and as
a mathematician.

Indeed, the macho competitiveness of much of pure mathematics struck me very
strongly when I was an undergraduate student: I switched then to
mathematical statistics because the teachers and students in that discipline
were much less competitive towards one another. For a long time, I thought
I was alone in this view, but I have since heard the same story from other
people, including some prominent mathematicians. I know one famous category
theorist who switched from analysis as a graduate student because the people
there were too competitive, while the category theory people were more
co-operative.

It may be worth mentioning that I am male. In other words, a dislike of competitiveness is not confined to women. The
statistics department I entered as an undergraduate, for example, had no
women in it, yet was much less competitive than the pure mathematics
department (which had once been headed by a woman). I think it is
disciplinary tradition rather than gender that is the key factor here.

I am now a Computer Scientist. I have also found differences in the competitiveness of people in different sub-domains of CS.
To generalize greatly, I have found people in Artificial Intelligence (AI)
much less macho and competitive than those in (say) Algorithm and Complexity
Theory. Within AI, people in (say) Argumentation are generally much less
macho and competitive than those in Game Theory and Mechanism Design. In each
case, the more formal and mathematical the domain, the more competitive it
tends to be. It could be that these domains have acquired their cultures
from mathematicians, while the other domains have been less influenced by
the culture of mathematics.

From factory workers to Wall Street bankers, a reasonable proficiency in math is a crucial requirement for most well-paying jobs in a modern economy. Yet, over the past 30 years, mathematics achievement of U.S. high school students has remained stagnant — and significantly behind many other countries, including China, Japan, Finland, the Netherlands and Canada.

A research team led by Carnegie Mellon University’s Robert Siegler has identified a major source of the gap — U. S. students’ inadequate knowledge of fractions and division. Although fractions and division are taught in elementary school, even many college students have poor knowledge of them. The research team found that fifth graders’ understanding of fractions and division predicted high school students’ knowledge of algebra and overall math achievement, even after statistically controlling for parents’ education and income and for the children’s own age, gender, I.Q., reading comprehension, working memory, and knowledge of whole number addition, subtraction and multiplication. Published in Psychological Science, a journal of the Association for Psychological Science, the findings demonstrate an immediate need to improve teaching and learning of fractions and division.

“We suspected that early knowledge in these areas was absolutely crucial to later learning of more advanced mathematics, but did not have any evidence until now,” said Siegler, the Teresa Heinz Professor of Cognitive Psychology at Carnegie Mellon. “The clear message is that we need to improve instruction in long division and fractions, which will require helping teachers to gain a deeper understanding of the concepts that underlie these mathematical operations. At present, many teachers lack this understanding. Because mastery of fractions, ratios and proportions is necessary in a high percentage of contemporary occupations, we need to start making these improvements now.”

The research, supported by grants from the U.S. Department of Education’s Institute of Education Sciences and by the National Science Foundation’s Developmental and Learning Science Group at the Social, Behavioral, and Economic Directorate, was conducted by a team of eight investigators: Siegler; U.C. Irvine’s Greg J. Duncan; the University of Michigan’s Pamela E. Davis-Kean, Maria Ines Susperreguy and Meichu Chen; the University of London’s Kathryn Duckworth; the University of Chicago’s Amy Claessens; and Vanderbilt University’s Mimi Engel.

For the study, the team examined two nationally representative data sets, one from the U.S. and one from the United Kingdom. The U.S. set included 599 children who were tested in 1997 as 10-12 year-olds and again in 2002 as 15-17-year-olds. The set from the U.K. included 3,677 children who were tested in 1980 as 10-year-olds and in 1986 as 16-year-olds. The importance of fractions and division for long-term mathematics learning was evident in both data sets, despite the data being collected in two different countries almost 20 years apart.

“This research is a good demonstration of what collaborations between psychologists, economists, public policy analysts and education scientists can create,” said Davis-Kean, associate professor of psychology at Michigan. “Instead of relying on results from a single study, this study replicates findings across two national data sets in two different countries, which strengthens our confidence in the results.”

Rob Ochsendorf, program officer for special education research at the U.S. Department of Education’s Institute for Special Education Research added, “This study is critical for providing empirical and general confirmation of the crucial role of division and fractions proficiency for long-term success in mathematics for all students. The results provide important cues to educators and researchers regarding the skills that are ripe for intervention in order to improve overall mathematics achievement in the U.S.”